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Sep 19th, 2019
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  1. % i
  2. clc;
  3. clear;
  4. tspan = [0,10];      % Visualise solution from 0 to 5 seconds
  5. x0 = [1,1];             % Initial condition of 1
  6.  
  7. L = 3;
  8. kappa = 2;
  9. rho = L + kappa/sqrt(2);
  10. sigma = @(x,z) x^2 + 4*x - 0.5*z;
  11. v = @(x,z) -rho*sign(sigma(x,z));
  12. u = @(x,z) v(x,z) - 2*x^3 - 10*x^2 - 12*x + x*z + 2*z;
  13.  
  14. [t,x] = ode45(@(t,x) f_1(x,t,u),tspan,x0);
  15.  
  16. x_surface = linspace(0,2,10);
  17. z_surface = 2*x_surface.^2 + 8*x_surface;
  18.  
  19. sigma_val = zeros(length(x),1);
  20. u_val = zeros(length(x),1);
  21. for i = 1:length(x)
  22.     sigma_val(i) = sigma(x(i,1),x(i,2));
  23.     u_val(i) = u(x(i,1),x(i,2));
  24. end
  25.  
  26. % Visualise solutions
  27. figure('Name','i_1','NumberTitle','off','Position', [100, 100, 700, 400]);
  28. plot(x(:,1),x(:,2),'Linewidth',2);
  29. hold on
  30. plot(x_surface,z_surface,'Linewidth',2);
  31. title('x(t), z(t) With sliding surface \sigma = 0');
  32. xlabel('x(t)');
  33. ylabel('z(t)');
  34.  
  35. figure('Name','i_2','NumberTitle','off','Position', [100, 100, 700, 400]);
  36. subplot(1,3,1);
  37. plot(t,x(:,1),'Linewidth',2);
  38. title('t, x(t)');
  39. xlabel('Time (s)');
  40. ylabel('x(t)');
  41. xlim(tspan);
  42.  
  43. subplot(1,3,2);
  44. plot(t,x(:,2),'Linewidth',2);
  45. title('t, z(t)');
  46. xlabel('Time (s)');
  47. ylabel('z(t)');
  48. xlim(tspan);
  49.  
  50. subplot(1,3,3);
  51. plot(t,sigma_val,'Linewidth',2);
  52. title('t, \sigma(t)');
  53. xlabel('Time (s)');
  54. ylabel('\sigma(t)');
  55. xlim(tspan);
  56.  
  57. figure('Name','i_3','NumberTitle','off','Position', [100, 100, 700, 400]);
  58. plot(t,u_val,'Linewidth',2);
  59. title('t, u_{sm}(t)');
  60. xlabel('Time (s)');
  61. ylabel('u_{sm}(t)');
  62. xlim(tspan);
  63.  
  64. %% ii
  65. clear x z
  66. epsilon = 0.1;
  67. v = @(x,z) -rho*min(max((sigma(x,z)/epsilon),-1),1);
  68. u = @(x,z) v(x,z) - 2*x^3 - 10*x^2 - 12*x + x*z + 2*z;
  69. [t,x] = ode45(@(t,x) f_1(x,t,u),tspan,x0);
  70. sigma_val = zeros(length(x),1);
  71. u_val = zeros(length(x),1);
  72. for i = 1:length(x)
  73.     sigma_val(i) = sigma(x(i,1),x(i,2));
  74.     u_val(i) = u(x(i,1),x(i,2));
  75. end
  76.  
  77. % Visualise solutions
  78. figure('Name','ii_1','NumberTitle','off','Position', [100, 100, 700, 400]);
  79. subplot(1,3,1);
  80. plot(t,x(:,1),'Linewidth',2);
  81. title('t, x(t)');
  82. xlabel('Time (s)');
  83. ylabel('x(t)');
  84. xlim(tspan);
  85.  
  86. subplot(1,3,2);
  87. plot(t,x(:,2),'Linewidth',2);
  88. title('t, z(t)');
  89. xlabel('Time (s)');
  90. ylabel('z(t)');
  91. xlim(tspan);
  92.  
  93. subplot(1,3,3);
  94. plot(t,sigma_val,'Linewidth',2);
  95. title('t, \sigma(t)');
  96. xlabel('Time (s)');
  97. ylabel('\sigma(t)');
  98. xlim(tspan);
  99.  
  100. figure('Name','ii_2','NumberTitle','off','Position', [100, 100, 700, 400]);
  101. plot(t,u_val,'Linewidth',2);
  102. title('t, u_\epsilon(t)');
  103. xlabel('Time (s)');
  104. ylabel('u_\epsilon(t)');
  105. xlim(tspan);
  106.  
  107. %%
  108. function dX = f_1(x,t,u)
  109. dx = x(1)^2 + 3*x(1) - 0.5*x(2);
  110. dz = -2*u(x(1),x(2)) + cos(t) + 2*sin(3*x(1)*x(2));
  111. dX = [dx;dz];
  112. end
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